There are two types of Audit sampling

Define audit sampling:

is a selection of less than 100% of the population of audit relevance, and the evaluation of that sampel, such that the selected items will be representative of the Pop.

a rep. sample is one that will result in conclusions that, subject to the limitations of sampling risk, are similiar to what would have been if the whole pop was tested.

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AICPA Audit and ACcounting Guide, Audit Sampling, lists procedures that do not invole sampling as follows:

1) Inquiry and obersvation2) anlytical procedures3) procedures applied to every item in the pop.4) tests of contorls where application is not documented5) procedures from which the auditor does not intend to extend a conclusion to the remaining items in the account.6) untested balnaces

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Audit sampling - nonstatistical and statistical

a) both involve judgment in planning, executing the mspaling plan, and eval. the results.b) both can provide suficent appropriate audit evidencec) statistcal sampling helps the auditor to:1) design an efficient smaple2) measure the sufficiency of the audit evidence obtained.3) evaluate the sample results. -auditor cna qnatify smapling risk to a limitit to a level considerd acceptable.

is a combination of the risk that a material misstatement will occur and not be detected.

AR = IR X CR x DR

Nonsampling risk

incldues all aspects of audit risk that are not due to sampling. It is controlled by adequate planning and supervision of aduit work and proper adherence to quality contorl standards. Ex.1) the failure to select appropriate audit procedures2) The failure to recognize missatements in documents ex.3) misinterpreting the results of audit tests.

Think of human type errors

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Sampling risk:

is the risk that auditor's conclusion, based on a sample, might be different from the conclusion that would be reached if the test wer eapplied to the entire thing.

A) tests of contorls smapling risk include risk of assesing control risk too high and control risk too low.B) Substantive test sampling risks including the risk of incorrect rejection and the risk of incorrect acceptance.

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Types of Audit Tests in which sampling may be used

A) Test of controls B) Substantive TestsC) Dual purpose tests - when chosen, auditors select the sample size as the higher of that required for the two purposes. Ex. if the test of control test required 45 items, and the substantive test required 40, both tests would be perforemd using the 45 items.

is the risk that the assessed level of control risk based on the sample is greater than the true operating effectiveness of the contorl structure policy or procedure. This risk relates to AUDIT EFFICIENCY. If the audtior assess contorl risk too high, substantive tests will consequently be expended beyong the ncessary level, leading to audit inefficency.

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Risk of assesing control risk too low (beta risk, type II error)

is the risk that the assessed level of control risk based on the sample is less than the true opearting effectiveness of the contorl structure policy or procedure. Know that his risk relates to AUDIT EFFECTIVENESS. If the auditor asesses contorl risk too low, subsatntive tests will not be expanded to the ncessary level to ensure effective audit.

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Statistical (Attributes) sampling for Test of Controls

A. Steps involved in attributes sampling.1. Determine the objectives of the test - remember that tests of contorls are designed to provide reasonable assurance that internal contorl is operating effectively. For ex., attirubtes sampling might test contorls for voucher procesing, billing sytmes. etc.

Uusally use when there's a trail of documentary evidence.

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Define the deviation conditions -

An auditor should identify characteristics (attributes) that would indicate opeartion of the internal control procedures. The auditro next defines the possible deviation conditions. A divation is a depature from the prescribed IC policy and procedure.

ex. If the prescribed procedure to be tested requires the cancellation of each paid voucher, a paid but uncanceled voucher would constitute a deviation.

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Define a pop. -

For tests of controls, the population is the class of tranactions being tested. Conclusions based on sample results can be projected only to the pop. from which the smaple was seleceted.

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Three steps to defining a pop.

1) Define the period covered by the test- ideally, tests of controls should be applied to tranactions executed during the netier period under audit. In some cases it is more efficient to test tranactions at an interim date and use supplemental procedures to obtain reasonble assurance regarding the remaining period. 2) Define the sampling unit - the sampling unit is one of the indivudal elements constituting the population. IN our earlier example, the sampling unit si the voucher.3) Consider the completeness of the population - the auditor actually selects sampling units

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Method to selecting the sample -

1) Random number sampling - every sampling unit has the same probability of being selected, and every combination of sampling units of equal size has the same probability of being selected. Ranodm numbers can be generated.2) Systematic sampling - every nth (pop. size/sample size) item is selected after a random start. When a random starting point is used, this method provides every sampling unit in the pop an equal cahnce of being selected.3) Haphazard sampling - a sample consisting of units selected without any conscious bias, that is, without any special reason for including or omitting items form the smaple. 4) block sampling 0 a sample consisting of a group of units.

Determine the sample size

1. Risk levels between 1% and 10% are normally used.2) There is an inverse relationship (e.g., as one increases the other decreases) between the risk of asessing contorl risk too low and smaple size.

B. Tolerable rate - The maximum rate of deviation from a prescribed control structure policy or procedure that an auditor is willing to accept without modifying the planned assessed level of contorl risk.

1. The auditor's detrmination of the tolerable deviation rate is a function of a) the palnned assesed level of control risk.b) the degree of assurance desired by the sample.

2) When the adutiro's planned assessed level of cnotrol risk is low, and degree of assurance desired from the smaple is high, the tolerable rate shoudl be low.

A. this will be the case for ex., when the audtiro does not perform other tests of control.

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C. Expected pop deviation rate (expected rate of occurence)- an estimate of the deviation rate in the entire pop.

1) If expeceted pop. deviation rate exceeds the tolerable rate, tests will not be performed.2) Although the risk of assessing control risk too high is often not explicitly controlled when determining attributes sample size, it can be controlled to some extent.3) There is a direct relationsihp between the expected deviation rate and sample size. 4) the expected pop. deviation rate is typically determined by using last year's deviation rate adjusted judgmentally for curent year changes in contorl proceudres.b. used only to detmeirne smaple size and not to evaluate sample results. 5) Deviation from a contorl procedure does not ncessarily result in a missatement, the rate of missatements is generally lower than the deviation rate.

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Population effect

Increases in the size of the pop. nromally increase the sample size. However, it is generally appropriate to treat any pop. of more than 5000 sample units as if it were infinite.

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Fixed vs. sequential sample size approach

Audit samples may be designed using either a fixed or a sequential sample size approach. Supplmenting traditional attributes (fixed size) sampling approaches are1) Sequential (stop or go) sampling - a sampling plan or which the sample is selected in several steps, with the need to perform each step conditional on the results of the preivous steps. That is, the results may either be so poor as to indicate that the contorl may not be relied upon, or so good as to justify reliance at each step.2) Discovery sampling - a procedure for determining the sample size required to have a stipulated probability of observing at least one occurrence when the expected pop. deviation rate is at a deisgnated level. Most appropriate hwen expected deviation rate is zero or near zero. If deviation is detected, the auditor must either 1 ) use an alternate approach, 2 ) if the deviation is of sufficent importance, audit all tranactions.

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Perform the sampling plan:

Each deviation should be analyzed to detmerine whether it is an isolated or recurring type of occurence.

A) The auditor should select extra sample items so that voided, unused, or inaplicable documents can be excluded.B) If the auditor is unable to examine a selected item, it should be consdiered a deviation from evaluation purposes. Consider the reasons for the limitation and it's implications.C) If early indication that control cannot be relied upon, no need to continue the test.

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Evaluate the sample results

Once audit procedures have been perfoermed on all sample items, the sample results must be evaluated and proejcted to the entire pop. from which sample was selected.

a.To calculate sample deviation rate: 8/150 = 5.3%b. The upper deviation limit found from the table for a 5% risk of assessing control risk too low and 8 deviations = 9.5%c. The allowance for sampling risk= 9.2-5.3=4.2%d. the conclusions that can be drawn includei. there is a 5% cahnce of the true pop. deviation rate being greater than or equal to 9.5%.ii. the upper deviation limit 9.5% exceeds the tolerable deviation rate 6%, the smaple results indicate that contorl risk for hte contorl procedure being tesed is higher than planned, and therefore, the scope of resulting substantive tests must be increased.

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The auditor should consider the qualitative aspects of each deviation:

1) nature an cuase of each deviation should be analyzed.2) the possible relationship of the deviations to other phases of hte aduit.

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Conclusion on sampling results:

1) If all evidence obtained, including sample results, supports the aduitor's planned assessed level of contorl risk, the auditor generally does not need to modify planned substnative tests.2. If the planned level is not supported, the auditor willa. test other related controls.b. modify the related substantive tests to relfect increased control assesment.

1) In nonstatistical sampling it is not ncessary to quantify these factors.2. the auditor should still consider the effects on sample size.

b. sample slection. - any sample method.c. evaluate sample reuslts.

1) In nonstatistical sampling it is impossible to determine an upper deviation limit or to quantify sampling risk.

The auditor should relate the deviation rate in a the sample to the tolerable rate established in the deisng stage to determine wheter an adequate allownace for smapling risk has been provided to draw the conlusion that acceptably low level of risk.a. rule of thumb 0 if the deviaotn rate in a properly deisnged smaple does not exceed the expected pop. deviation rate used in determing sample size, then it's OK!2. As in statistical smapling, the qualatative aspects of devaitons should be consdierd in adition to the frequency of devaitions.3) again the audtior must use his/her professional judgement to reach an overal lconslucions.

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Sampling in Substantive tests of details

Determine the objectives of the test -

Variable sampling is used primarily for substantive testing and that its conclusion is generally stated in dollar terms. Varible sampling might test, the recorded amount of accounts rreceivable, payroll expense, etc.

Consider the ocmpleness of the pop. 0 is the physical represtnation incldues the entire pop?

substantive tests of details - select an audit sampling technique

substantive tests of details - 5 items need to be considerd

1) variation within the pop.2) acceptable level fo risk3) tolerable missatment - an estimate of the maximum monetary missatement that may exist in an account balance or class of tranactions, which combined with missatmenets in other acocunts, without causing the FS to be materially missated.4) expecetd amount of missatmenet - expected missatement is estimated using an unverstanding of the business, prior year info, a pilot smaple, or the results of the review.5) pop. size - sample size increass as pop size incease.

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Probaiblity proportional to size (PPS) sampling

Sampling interval = book value of pop. / sample size

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Projected misstatement

is calculated for each logical unit containing misstaemnets and totaled. The approach differes based on whether the book value of the specific logical unit is less than or great than the size of hte sampling interval. When logical unit is less than smapling interval.Taint value is the residual value.( misstatement/sample). The taint value is then multiple by the sampling interval.

When logical unit greater than sampling interval. - the actual maount of missatement is considered to be the projected misstatement. For ex. BV of 4k, an audit value of 40, and a sampling interval of 3000. The projected missatemnt is 3960. Simiarly if the audlt value is 3300, then the projecetd missatemnt is 700.

The various projected missatments are totaled if there is more than one account.

1) If the upper limit on missamtents is less than or equal to tolerable misstatement, the sample results suport the conclusion that the pop. is not missated by more than tolerable misstatement at the specified risk of incorrect acceptance.2) If hte upper limit on missatments is greater than the tolerable missatmtent, the sample results do not support the conclusion that the pop. is not missated by more than the tolerable missatment. This may be due to the fact that A) the op. is misstated. B) auditor's expectation of missatment was low or resutled in too small of a smpale3) The auditor should consider qualitative aspects of errors found as well as quantitative factors.

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Observation:

if no missatements are found, projected missatemnet and the incremental allowance for projected missaments will equal zero, leaving the basic precision as teh only nonezero.

Basic precision

Reliability factor x sampling interval

what

Classical Variables Salmping

- uses normal distrubtion theory to evaluate selected characteristics of a pop. on the basis of a sample fo the items constituting the pop.

For large samples (greater than or equal to 30), the distribution of sample means tends to be normally distributed about its own mean which is equal to the true pop. mean,

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variations of classical variables samlping.

Mean-per-unit estimation:

is a classical variables sampling technique that proejcts the sample average to the total pop. by multiplying the sample average by the number of items in the pop.A. Determine audit values for each sample item.B. Calculate the average audit amount.C. Multiply this average audit amount times the number of units in the pop. to obtain the estimated pop. value.

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Difference estimation

is a calssical variables sampling technique that uses the average difference between audited amounts and individual recorded amounts to estimate the total audited amount of a pop. and an allowance for sampling risk.

A. Determine audit values for each sample item.B. Calculate the difference between the audit value and book value for each.C. Calvaulate the average diference.D. Determine the estimated pop. value by multiplying the average difference by the total pop. units and adding or subtracting this value from the recorded book value.

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Ratio estimation

classical variable sampling technique that uses the ratio of audited amounts to recorded amounts.

It is used when the calculated audit amounts are approximately protional to the client's book amounts.

When there are is numerous pricing nad extension errors.

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Comparions of PPS sampling to classical sampling. Advtages of PPS over smapling

(1) Generally easier to use.(2) Size of sample not based on variation of audited amounts.(3) Automatically results in a stratified sample.(4) Individually significant items are automatically identified.(5) Usually results in a smaller sample size if no misstatements are expected.(6) Can be easily designed and sample selection can begin before the complete population is available.

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Advantages of classical variables samlping over PPS

(1) May result in a smaller sample size if there are many differences between audited and book values.(2) Easier to expand sample size if that becomes necessary.(3) Selection of zero balances does not require special sample design considerations.(4) Inclusion of negative balances does not require special sample design considerations.

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Effectiveness of the audit:!

the risks of incorrect acceptance and assessing control risk too low relate to the effectiveness of an audit in detecting an existing material misstatement.

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for an auditor to estimate the appropriate sample size, the auditor must first know

the acceptable level of risk.

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deviation rate:

applies to attribute sampling which is appropriate for tests of controls, not substantive tests

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When using classical variables sampling for estimation, an auditor normally evaluates the sampling results by calculating the possible error in either direction. This statistical concept is known as

Precision.

precision (the allowance for sampling risk) is used to construct an interval around the sample results in which the true population characteristic is expected to lie. Accordingly, possible error in either direction from the sample results is considered.

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Discovery sampling.

is a type of attribute sampling used in tests of controls when very low occurrence rates are expected and the auditor wants a chance to observe one occurrence.

Use this IF the auditor is conscnered that a pop. may contain exceptions, the determination of a sample size sufficient to include a tleast on such exception.

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The major reason that the difference and ratio estimation methods would be expected to produce audit efficiency is that the

Variability of the populations of differences or ratios is less than that of the populations of book values or audited values.

This answer is correct because difference and ratio estimation methods measure the difference between audit and book values or the ratio of audit to book values. As these differences should not be great, the population of these differences will have little variance. In statistical sampling the less variation in a population, the smaller the required sample to provide an estimate of the population. In other words, difference and ratio estimation methods are more efficient because the differences between audit and book values are expected to vary less than the actual items in the population.

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As a result of sampling procedures applied as tests of controls, an auditor incorrectly assesses control risk lower than appropriate. The most likely explanation for this situation is that

The deviation rate in the auditor’s sample is less than the tolerable rate, but the deviation rate in the population exceeds the tolerable rate.

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the ratio estimation technique:

the ratio estimation technique estimates the audited value by multiplying the ratio of the audited value over the book value of the sample by the population book value. In this case, ($300,000/$250,000) × $5,000,000 = $6,000,000.

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The major reason that the difference and ratio estimation methods would be expected to produce audit efficiency is that the

Variability of the populations of differences or ratios is less than that of the populations of book values or audited values.

This answer is correct because difference and ratio estimation methods measure the difference between audit and book values or the ratio of audit to book values. As these differences should not be great, the population of these differences will have little variance. In statistical sampling the less variation in a population, the smaller the required sample to provide an estimate of the population. In other words, difference and ratio estimation methods are more efficient because the differences between audit and book values are expected to vary less than the actual items in the population.

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the tolerable rate of deviation is

an input into determination of sample size (the others include expected deviation rate, the risk of assessing control risk too low, and population size [which has only a very limited effect on sample size]).

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Which of the following would be a consideration in planning an auditor’s sample for a test of controls?

auditors use attributes sampling for tests of controls, and to use attributes sampling, auditors need the allowable risk of assessing control risk too low, the expected population deviation rate, and the tolerable rate.

In planning a statistical sample for a test of controls, an auditor increased the expected population deviation rate from the prior year’s rate because of the results of the prior year’s tests of controls and the overall control environment. The auditor most likely would then increase the planned

This answer is correct because an increased population deviation rate will require an increase in sample size when the planned assessed level of control risk is to remain the same.

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In performing tests of controls, the auditor will normally find that

The rate of deviations in the sample exceeds the rate of error in the accounting records.

This answer is correct because the existence of a deviation from a control activity may or may not indicate that an error exists in the accounting records. For example, an invoice which has not been footed to test mathematical accuracy may still be correctly totaled and properly recorded in the accounting records.

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the primary objective of probability-proportional-to-size sampling?

best at concluding on overstatements as contrasted to understatements; this is due to characteristics of the underlying statistical theory.

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what is Incorrect acceptance?

An auditor discovers that an account balance believed not to be materially misstated based on an audit sample was materially misstated based on the total population of the account balance. This is a sampling types of risks.

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auditors use attributes sampling for tests of controls, and to use attributes sampling, auditors need the

allowable risk of assessing control risk too low, the expected population deviation rate, and the tolerable rate.

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What is incorrect rejection?

Is the risk that the smaple supports the conclusion that the recorded account balance is materially missated when the account is not misstated.

The risk fo assessing control risk too high or the risk fo assessing control risk too low relate to tests of contorls, and not to substantive tests of details. SO REMEMBER? are you testing controls or substantive testing?

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The type of sampling plan that does not use a fixed sample size for tests of controls?

Sequential sampling ersults in the slection of a sample in several steps, with each step conditional on the result of the previous steps. Therefore, sample size will vary depending upon the number of stages that provde necessary.

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What is stratified sampling?

is a technique of breaking the pop. down into subpopulations and sapplying different smaple selection methods to subpopulations. Used to minimize the variance within the overall pop.

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What factos are consdiered in determinig the sample size for test of controls?

known misstatement and likely misstatements

AU 312.08 defines known misstatements as “specific misstatements identified during the audit arising from the incorrect selection or misapplication of accounting principles or misstatements of facts identified, including for example, those arising from mistakes in gathering or processing data and the overlooking or misinterpretation of facts.”

It further defines likely misstatements as “misstatements that (a) arise from differences between management's and the auditor's judgments concerning accounting estimates that the auditor considers unreasonable or inappropriate and (b) the auditor considers likely to exist based on an extrapolation from audit evidence obtained.”

An unrecorded liability resulting from a specific activity or invoice would be considered a known misstatement. The other two answer choices are examples of likely misstatements.